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Rare Kaon Decays in the 1/Nc-Expansion

arXiv:hep-ph/9209231v1 14 Sep 1992CPT-92/P.2795Rare Kaon Decays in the 1/Nc-ExpansionC. Bruno and J. PradesCentre de Physique Th´eorique, C.N.R.S.

- Luminy, Case 907F-13288 Marseille Cedex 9, FranceAbstractWe study the unknown coupling constants that appear at O(p4) in the Chiral Per-turbation Theory analysis of K →πγ∗→πl+l−, K+−→π+−γγ [1] and K →ππγ[2] decays. To that end, we compute the chiral realization of the ∆S = 1 Hamilto-nian in the framework of the 1/Nc-expansion of the low-energy action proposed inRef.

[3]. The phenomenological implications are also discussed.CPT-92/P.2795July 1992

1IntroductionIn this work we shall study the unknown coupling constants that govern the anal-ysis of K →πγ∗→πl+l−, K+−→π+−γγ [1] and K →ππγ [2] decays at O(p4) inthe framework of Chiral Perturbation Theory (ChPT). We shall restrict ourselvesto the non-anomalous sector of the theory.

(For a recent discussion of anomalousnon-leptonic decays see Ref. [2].) These transitions have become particularly in-teresting because of possible CP-violation effects in rare K-decays.The varioustests on this subject discussed in Ref.

[1] depend on the values of these couplingconstants. These are not fixed by chiral symmetry requirements alone but are, inprinciple, determined by the dynamics of the underlying theory.Recently there have been attempts to derive the low energy effective chiral ac-tion of QCD [4, 5] as well as some of the coupling constants of the effective chiralLagrangian of strangeness-changing four-quark Hamiltonian [3].

Here we shall cal-culate the above mentioned couplings in the framework of this approach.It is well known that gauge invariance together with chiral symmetry forbid theO(p2) contribution [1] to K-decays with at most one pion in the final state. The firstnon-vanishing contributions to the transition amplitudes K →πγ∗and K →πγγappear at O(p4).

The contributions from chiral loops have been calculated in Ref.[1]. The K →ππγ decays have been analysed in Ref.

[2] to the same order in thechiral expansion. In Refs.

[1, 6, 7], some of the couplings constants we are interestedin have been calculated within the context of various models.The effective non-leptonic chiral Lagrangian consistent with the octet structureof the dominant ∆S = 1 weak Hamiltonian has been classified in Refs. [6, 9].

ThisLagrangian, when restricted to the rare kaon decays processes we are interested incan be parametrized in terms of the set of coupling constants ω1, ω2, ω4, ω′1 and ω′2,introduced in Refs. [1, 2] as follows,L(4)eff = −GF√2 VudV ∗us f 2π4 g8hi ω1⟨f µν(+) {∆32 , ξµξν}⟩+ 2 i ω2⟨f µν(+)ξµ∆32ξν⟩+ 43 ω−−4⟨f µν(−)f(−)µν∆32⟩−4 ω++4⟨f µν(+)f(+)µν∆32⟩−2 ω−+4⟨nf µν(−) , f(+)µνo∆32⟩+ i ω′1⟨f µν(−) {∆32 , ξµξν}⟩+ 2 i ω′2⟨f µν(−)ξµ∆32ξν⟩i+ h.c.+ non −octet operators,(1)with ω4 = ω−−4+ ω++4+ ω−+4, where ⟨⟩denotes a trace in flavour.

The notationhere is defined below:1

f µν(±) = ξF µνL ξ† ± ξ†F µνR ξ,∆32 = ξλ32ξ†,ξµ = i ξ†DµUξ† = −i ξDµU†ξ,(2)where λab is the 3 × 3 flavour matrix (λab)ij = δaiδbj and ξ is chosen such thatU = ξ2. (3)U ≡exp−√2iΦfis a SU(3) matrix incorporating the octet of pseudoscalarmesons,Φ(x) ≡⃗λ√2 ⃗ϕ =π0√2 +η√6π+K+π−−π0√2 +η√6K0K−¯K0−2 η√6,(4)and f ≃fπ = 93.2 MeV is the pion decay constant at lowest order.

Dµ is a covariantderivative [4] which acts on U. In the presence of external electromagnetic fieldsonly, it is defined asDµ U ≡∂µ U −i|e| Aµ[Q, U],(5)where e is the electron charge and Q is a diagonal 3 × 3 matrix that takes intoaccount the electromagnetic (u,d,s)-light-quark charges, Q = 13diag(2, −1, −1).

Inthis case we also haveF µνL= F µνR= |e|QF µν = |e|Q(∂µAν −∂νAµ). (6)In Eq.

(1) GF is the Fermi constant and Vij are the Cabibbo-Kobayashi-Maskawa(CKM) matrix elements. The g8-coupling introduced in Eq.

(1) is the constant thatmodulates the octet operator in the ∆S = 1 effective Lagrangian of order p2; i.e.,L(2)eff = −GF√2 Vud V ∗us f 4π g8 ⟨∆32 ξµξµ⟩+ h.c. + non −octet operators. (7)We want to perform a calculation of the coupling constants which appear in L(4)effin Eq.

(1) in the context of an expansion in powers of 1/Nc, (Nc = number ofcolours). The paper is organized as follows.

In Section 2 we shall introduce the∆S = 1 effective Hamiltonian. In Section 3 we shall study the large-Nc limit resultfor the counterterms defined in Eq.

(1). In Section 4, the effective realization of the2

∆S = 1 four-fermionic Hamiltonian is studied including terms of O(Nc(αsNc)) asin Ref. [3].

Phenomenological implications of our results are presented in Section 5and finally, in Section 6, the conclusions are given.2The ∆S = 1 effective HamiltonianThe ∆S = 1 chiral Lagrangians in Eq. (1) and (7) are part of the effectiverealization of the corresponding Standard Model (SM) sector in terms of low-energydegrees of freedom.

In the SM with three flavours, once the heaviest particles (W-boson, t-, b- and c-quarks) have been integrated out, the complete basis of operatorsof weak and strong origin that induce strangeness changing processes with ∆S = 1via W-exchange is given by,Q1 = 4(¯sLγµdL)(¯uLγµuL)Q2 = 4(¯sLγµuL)(¯uLγµdL)Q3 = 4(¯sLγµdL)Xq=u,d,s(¯qLγµqL)Q4 = 4Xq=u,d,s(¯sLγµqL)(¯qLγµdL)Q5 = 4(¯sLγµdL)Xq=u,d,s(¯qRγµqR)Q6 = −8Xq=u,d,s(¯sLqR)(¯qRdL). (8)where ΨRL ≡12 (1±γ5) Ψ(x), and summation over colour indices is understood insideeach bracket.The reduction of the SM Lagrangian to an effective electroweak Hamiltonian hasbeen discussed in the literature [11].The structure of the effective non-leptoniclow-energy Hamiltonian is the following,H∆S=1eff= GF√2 VudV ∗usn12C+(µ2) (Q2 + Q1) + 12C−(µ2) (Q2 −Q1)+ C3(µ2) Q3 + C4(µ2) (Q3 + Q2 −Q1) + C5(µ2) Q5 + C6(µ2) Q6} + h.c.(9)The Wilson coefficients C± and Ci, i = 3, 4, 5, 6 are known functions of the heavymasses and the renormalization scale µ beyond the leading logarithmic approxima-tion [12].

In the limit we are working here we shall use these coefficients in theleading logarithmic approximation. Of course, the matrix elements of the Qi=1,···,6operators must depend on the µ scale in such a way that physical amplitudes do notdepend on it.

We shall be only interested in the octet components of the four-quark3

operators that induce ∆I = 1/2 transitions (due to the octet dominance of the∆I = 1/2 enhancement); namely, Q−≡Q2 −Q1, the octet part of Q+ ≡Q2 + Q1and Q3, Q4, Q5, Q6 which are pure octet operators.To these ∆S = 1 operators in (8) of weak and strong origin we have to addtwo more operators that induce ∆S = 1 transitions, coming from the so-calledelectroweak penguins [13],QV7 = e22π (¯sLγµdL) ¯lγµlandQA7 = e22π (¯sLγµdL) ¯lγµγ5l,(10)with their corresponding Wilson coefficients, CV7 (µ2) and CA7 (µ2). So the matrix ele-ments of the Qi=1,···,6 operators have to be evaluated to order α = e2/4π whereas thematrix element of QA,V7must be calculated to order α0.

The effective Hamiltonianfor QA,V7can be written asHQ7eff = GF√2 VudV ∗usnCV7 (µ2) QV7 + CA7 (µ2) QA7o+ h.c.(11)The Wilson coefficient CA7 only receives contributions from the so-called Z penguinand W box diagrams. In the present paper we are just interested in transitionsthat are mediated by external photons, therefore we are not going to consider theelectroweak penguin operator modulated by CA7 and we have CV7= Cγ7 .Theexpression for the electromagnetic penguin Wilson coefficient Cγ7 (µ2) can be foundin Ref.

[13]. This Wilson coefficient contains a complex phase coming from theCKM matrix elements that can give rise to “direct” CP-violation effects.

In the restof the paper when we refer to the Q7 operator we would mean Qγ7.The other two ∆S = 1 operators which arise from electromagnetic penguin-likediagrams, see Ref. [14], start to contribute at order α2.3The ∆S = 1 effective Hamiltonian in the large-Nc limitIn the large-Nc limit, the ∆S = 1 effective Hamiltonian is given by,H∆S=1eff= GF√2 VudV ∗us Q2 + h.c..(12)In this limit we just need to calculate the effective action for the factorizable patternof the four-quark operator Q2.

This can be performed in a model independent wayby doing appropriate products of the low-energy realization of quark currents in theframework of effective chiral Lagrangians.4

To obtain the large-Nc limit effective realization at O(p4) we need to know theeffective realization of quark currents up to O(p3).These can be easily derivedfrom the O(p2) and O(p4) strong effective chiral Lagrangian given in Refs. [8, 10]in the presence of external sources.

In addition, we want to keep only the octetcomponent of the isospin-1/2 operators. In this limit the value of the g8-couplingin Eq.

(7) is (g8)1/Nc = 3/5, to be compared with the experimental value fromK →ππ decays, |g8|exp ≃5.1. For the coupling constants ω1,2,4 and ω′1,2 of theO(p4) effective Lagrangian in (1) we then obtain the following results,ω1 = ω2,g8 ω2 = 8 (g8)1/Nc L9,g8 ω4 = 12 (g8)1/Nc L10,g8 ω′1 = 8 (g8)1/Nc (L9 −2 L10),andω′2 = 0.

(13)Here L9 and L10 are two of the couplings that appear modulating local terms in thechiral Lagrangian at order p4. In the notation of Gasser and Leutwyler [8], theyread as follows,−i L9 ⟨F µνL DµUDνU† + F µνR DµU†DνU⟩+ L10⟨UF µνR U†FL,µν⟩.

(14)The present experimental results on the K+ →π+e+e−process [15] allow for thedetermination of the combination of coupling constants,g8(ω1 + 2 ω2) = 0.41+0.10−0.05. (15)Numerically, in the large-Nc limit we findg8(ω1 + 2 ω2) = 24 (g8)1/Nc L9 = 0.10;(16)i.e., a factor four lower than the experimental value.If one looks at Eq.

(13), the naive approach would be to consider that g8 factor-izes in the r.h.s. to all orders in the 1/Nc-expansion and therefore write down thefollowing result,ω1 = ω2 = 8 L9,ω4 = 12 L10,ω′1 = 8 (L9 −2 L10)andω′2 = 0.

(17)5

Thenω1 + 2 ω2 = 24 L9 = 0.16,(18)which is about twice the experimental value. In Section 4 we shall come back tothe question of whether or not factorization of g8 in Eq.

(13) remains valid afterincluding the next-to-leading corrections in the 1/Nc-expansion.It can be seen from the results above that we do not obtain the octet dominancerelation ω2 = 4 L9 which was assumed in Ref. [1].

Our results differ from thosefound in Ref. [16], where it is claimed the numerical relation ω1 = ω2 = 4 L9 inthe large-Nc limit.

In addition we get ω4 = 12 L10 which also differs from Ref. [16]where ω4 = 0.

The results found in Ref. [7] for ω1,2,4 from the “weak deformationmodel” are different to those we find in the large-Nc limit.The amplitude for the process K+−→π+−γγ to lowest non-trivial order inChPT was calculated in Ref.

[1]. The result turns out to depend on the followingcombination of coupling constants which is renormalization scale independent,ˆc = 32π2 h4(L9 + L10) −13(ω1 + 2 ω2 + 2 ω4)i= −32π23[(ω1 −ω2) + 3 (ω2 −4L9) + 2 (ω4 −6 L10)] .

(19)The combination L9 + L10 is a renormalization scale invariant quantity that isdetermined from the so-called structure term in π →eνγ [8, 17] to beL9 + L10 = (1.39 ± 0.38) × 10−3. (20)The combination ω1 + 2 ω2 + 2 ω4 is, of course, also renormalization scale invariant.In the large-Nc limit we find for the coupling constant ˆc the following value,ˆc = −128 π2 (g8)1/Ncg8(L9 + L10) = −1.8 (g8)1/Ncg8.

(21)Again, if we assume that factorization of g8 in the r.h.s. of Eq.

(13) is valid, wewould obtainˆc = −128 π2 (L9 + L10) = −1.8. (22)6

The “weak deformation model” of Ref. [7] predicts ˆc = 0.In the transition K+ →π+π0γ at O(p4) there appears the following combinationof coupling constants [2]: ω1 + 2 ω2 −ω′1 + 2 ω′2.

The combinations ω1 + 2 ω2 −ω′1 andω′2 are scale independent separately. For these combinations, we get the followingresults in the large-Nc limit:g8 (ω1 + 2 ω2 −ω′1) = 16 (g8)1/Nc (L9 + L10) = 0.01andω′2 = 0.

(23)In the following Section we shall estimate the next-to-leading corrections in the1/Nc-expansion to the couplings g8 ω1,2,4 and g8 ω′1,2 in the same spirit as the calcu-lation done for g8 in Ref. [3].4The effective action of four-quark operators at O(Nc(αsNc))In order to calculate the counterterms we are interested in, we shall need theeffective realization of the Qi=1,···,7 operators at O(p4).The procedure we shallfollow up is the one described in Ref.

[3] where the corresponding calculation toO(p2) has been reported. As there, we shall work in the chiral limit; i.e., we neglectthe (u,d,s)-light-quark masses.

Now, it is the complete set of strangeness changing∆S = 1 operators Qi=1,···,7 that has to be taken into account. Since we want to doour analysis in the context of the 1/Nc-expansion, it is worth to recall the behaviourin the large-Nc limit of the various parameters we have introduced: f 2π, L9, L10, ω1,2,4and ω′1,2 are order Nc; g8, C+, C−and C7 are order 1 and C3,4,5,6 are order 1/Nc.We want to perform a calculation of the effective Hamiltonian in (9) and (11) up toorder Nc(αsNc).

Only the Q+, Q−and Q7 operators are then needed at this order,since all the other four-quark operators are modulated by Wilson coefficients thatare already order 1/Nc. The Qi=1,···,5 operators and Qi=6,7 operators have differentspinorial structure, thus we are going to study them separately.

This will be donein Section 4.1 and Section 4.2, respectively. We shall summarize the correspondingresults for the couplings g8 ω1,2,4 and g8 ω′1,2 in Section 4.3.4.1The effective action of the Qi=1,...,5 operatorsAs it has been stated in the introduction of this Section, we need the effectiveaction of Q−and Q+ to order Nc(αs Nc); and to order N2c for the Q3,4,5 operators.7

The results to O(p4) for the terms which are relevant to the decays K →πγ∗,K+−→π+−γγ and K →ππγ are the following:⟨Q−⟩⇒−f 2π2 i L91 −g1Nc −γ−(µ) h⟨f µν(+) {∆32 , ξµξν}⟩+ 2 ⟨f µν(+) ξµ∆32ξν⟩i+ 4 L101 −g2Nc −γ−(µ)⟨f µν(−) f(−)µν ∆32⟩+ 2 iL91 −g1Nc −γ−(µ)−2 L101 −g2Nc −γ−(µ)⟨f µν(−) {∆32 , ξµξν}⟩+ 4 π (2 H1 + L10) αsNc (⟨Q6⟩+ ⟨Q4⟩) + 831 −g3Nc −γ−(µ)⟨Q7⟩;(24)and⟨Q+⟩⇒−f 2π52 i L91 + g1Nc −γ+(µ) h⟨f µν(+) {∆32 , ξµξν}⟩+ 2 ⟨f µν(+) ξµ∆32ξν⟩i+ 4 L101 + g2Nc −γ+(µ)⟨f µν(−) f(−)µν ∆32⟩+ 2 iL91 + g1Nc −γ+(µ)−2 L101 + g2Nc −γ+(µ)⟨f µν(−) {∆32 , ξµξν}⟩+ 4 π (2 H1 + L10) αsNc (⟨Q6⟩+ ⟨Q4⟩) −831 + g3Nc −γ+(µ)⟨Q7⟩+ non −octet terms. (25)The relevant terms for the effective action of the Q3, Q4, Q5 penguin operators are:⟨Q3⟩⇒O(Nc);(26)⟨Q4⟩⇒−f 2πn2 i L9h⟨f µν(+) {∆32 , ξµξν}⟩+ 2 ⟨f µν(+) ξµ∆32ξν⟩i+ 4 L10 ⟨f µν(−) f(−)µν ∆32⟩+ 2 i (L9 −2 L10) ⟨f µν(−) {∆32 , ξµξν}⟩o;(27)⟨Q5⟩⇒O(Nc).

(28)In the expressions above there appear three coupling constants of the chiral La-grangian at order p4 [8]. Two of them, L9 and L10, have been already introduced in8

Eq. (14).

The coupling constant H1 is the constant that modulates a contact termbetween external sources in the chiral Lagrangian at order p4 [8] as follows,H1 ⟨F µνR FR,µν + F µνL FL, µν⟩,(29)this constant is O(Nc) in the large-Nc limit. We have identified the coupling con-stants L9, L10 and H1 that appear in the effective action of Q−, Q+ and Q4 bycomparing our results with their respective expressions found in the context of themean-field approximation to the Nambu Jona-Lasinio model discussed in Ref.

[5].Thus, whenever their numerical values are needed, we shall use the values found inthis model.In Eqs. (24) and (25) we use the following short-hand notation for the leadingnon-perturbative gluonic corrections,g0 = 1 −12 G;g1 = 1 −16f 2π12M2QL9 G;g2 = 1 + 12f 2π24M2QL10 G;g3 = 1 + 13f 2π12M2Q(2H1 + L10) G,(30)withG ≡Nc ⟨αsπ G2⟩16π2f 4π,(31)which is O(1) in the large-Nc limit and the constituent quark mass MQ, which arisesfrom the following term,−MQ(¯qRUqL + ¯qLU†qR).

(32)This term is equivalent to the mean-field approximation of the Nambu Jona-Lasiniomechanism discussed in Ref. [5].

There are also perturbative gluonic correctionswhich we have gathered in the terms,γ−(µ) = −34αsπ ln µ2M2Q! ;γ+(µ) =34αsπ ln µ2M2Q!.

(33)9

From these expressions, it can be seen that the anomalous dimensions of the effectiveaction for the Q−and Q+ four-quark operators are the needed ones to compensatethe scale dependence of the Wilson coefficients.4.2The penguin operators Q6 and Q7Let us first calculate the effective action of Q6 since, as discussed in the intro-duction of this Section, we only need to know its result to leading O(N2c ). Thecalculation is rather straightforward and the result is⟨Q6⟩⇒−⟨ΨΨ⟩MQNc48π2h3 i ⟨f µν(+) {∆32 , ξµξν}⟩+ 2 i ⟨f µν(+) ξµ∆32ξν⟩+ 3 ⟨f µν(−) f(−)µν ∆32⟩+ 2 ⟨f µν(+) f(+)µν ∆32⟩i.

(34)Here ⟨ΨΨ⟩is a scale-dependent quantity which, at one-loop level, is defined by⟨ΨΨ⟩µ2 = 12 ln µ2Λ2MS!!4/9⟨c¯qq⟩. (35)where ⟨c¯qq⟩is the scale invariant quark vacuum condensate.

In the large-Nc limitthe quark condensate is order Nc.For the electromagnetic penguin operator Q7 we obtain the following effectiveaction,⟨Q7⟩⇒−332π f 2πh2 i ⟨f µν(+) {∆32 , ξµξν}⟩+ 2 i ⟨f µν(−) {∆32 , ξµξν}⟩−2 ⟨f µν(−) f(−)µν ∆32⟩−⟨nf µν(−) , f(+)µνo∆32⟩i,(36)which is valid to all orders in the 1/Nc-expansion.The terms in Eqs. (34) and (36) break the relation ω1 = ω2 which was found toleading O(N2c ).4.3ResultsThe coupling constants g8 ω1,2,4 and g8 ω′1,2 in Eq.

(1) can now be read offfromthe expression of the low-energy ∆S = 1 Hamiltonian, by inserting the effectiveaction of the four-quark operators Q−, Q+, Q3, Q4, Q5, Q6 and the electroweakpenguin operator Q7 that are in Eqs. (24)-(28) and (34)-(36), in Eqs.

(9) and (11).To evaluate these coupling constants, we have fixed the matching renormalization10

scale Λχ of the Wilson coefficients and the effective action of four-quark operators tobe of the order of the first vector meson resonance mass, (Λχ = Mρ = 770 MeV).We also need to know the coupling constant g8 defined in Eq. (7).

This coupling,within this framework was already computed in Ref. [3].

The result we get at theΛχ scale is the following,g8 = 12 C−(Λχ)1 −g0Nc −γ−(Λχ)+ 110 C+(Λχ)1 + g0Nc −γ+(Λχ)+ C4(Λχ) + 2πNc αs(Λχ) (2 H1 + L10) (C+(Λχ) + C−(Λχ))−16 L5 ⟨ΨΨ⟩2f 6πC6(Λχ) + 2πNc αs(Λχ) (2 H1 + L10)× (C+(Λχ) + C−(Λχ))+ O(αsNc). (37)We recall that L5 is one of the O(p4) constants needed to renormalize the UV-behaviour of the lowest order chiral loops [8].In the large-Nc limit L5 is orderNc.It turns out that the measurable quantities in the transitions we are interested inonly depend on the combinations: g8(ω1 −ω2), g8(ω2 −4L9), g8(ω4 −6L10) andg8(ω1 + 2 ω2 −ω′1 + 2 ω′2), [1, 2].

In the large-Nc limit the combination (ω1 −ω2)/L9is order 1/Nc whereas the combinations (ω2 −4 L9)/L9 and (ω4 −6 L10)/L10 areorder 1. From our calculation we find that the difference ω1 −ω2 depends only onthe penguin operators Q6 and Q7.At this point, it is worth coming back to the question of factorization of thecoupling constant g8 in the r.h.s of Eq.

(13).The expression for the effectiveaction of the Hamiltonian in Eqs. (9) and (11) calculated at order p4 in the chiralexpansion and at order Nc(αsNc) in the 1/Nc-expansion together with the expressionof the g8 coupling constant in Eq.

(37) lead us to the conclusion that the approachof factorizing out g8 in Eq. (13) is not valid when one considers next-to-leadingcorrections in the 1/Nc-expansion.

Therefore in the rest of the paper we shall giveresults for the combinations g8 ω1,2,4 and g8 ω′1,2.5Analysis of the resultsLet us now analyse some phenomenological implications which follow from ourcalculation of the coupling constants g8 ω1,2,4 and g8 ω′1,2. In Ref.

[1], the decayamplitudes of K →πγ∗were calculated at the one-loop level with the followingresults:11

A(K+ →π+γ∗) = GF√2 VudV ∗usg816π2 q2 bΦ+(q2) ǫµ(p + p′)µ ;(38)A(K0S →π0γ∗) = GF√2 VudV ∗usg816π2 q2 bΦS(q2) ǫµ(p + p′)µ ;(39)withbΦ+(q2) = 16π23[(ωr1 −ωr2) + 3(ωr2 −4Lr9)] (ν2)−ΦK(q2) + Φπ(q2) −13 lnmπmKν2;(40)bΦS(q2) = −16π23(ωr1 −ωr2) (ν2) + 2 ΦK(q2) −13 ln m2Kν2!. (41)Here,ΦK(π)(q2) = −4m2K(π)3q2+ 518 + 13 4m2K(π)q2−1!3/2arctan1/s4m2K(π)q2−1for q2 ≤4m2K(π).

(42)The coupling constants ω1, ω2, ω4, ω′1 and ω′2 are scale dependent quantities. In ourapproach, the explicit scale dependence of ω1,2,4 and ω′1,2 comes from next-to-leadingterms which we have not calculated.

We identify the values we get for ω1,2,4 andω′1,2 with those of ωr1,2,4 and ω′r1,2 renormalized at the ρ-resonance mass. We shallalso identify the constant Lr9 in Eq.

(40) with the coupling L9 renormalized at thissame scale. Following Ref.

[1] we define the constants,ω+ ≡−16π23[(ωr1 −ωr2) + 3(ωr2 −4Lr9)] (M2ρ) −13 ln mKmπM2ρ! ;(43)ωS ≡−16π23(ωr1 −ωr2) (M2ρ) −13 ln m2KM2ρ!.

(44)ThusbΦ+(q2) = −[ΦK(q2) + Φπ(q2) + ω+] ;(45)bΦS(q2) = 2ΦK(q2) + ωS;(46)and the decay rates for K →πl+l−can be written in the following way12

Γ(K →πl+l−) = ΓZ (1−√δ)24ǫdz λ3/2(1, z, δ)1 −4 ǫz1/2 1 + 2 ǫz|bΦ|2,(47)wherez =q2m2K, ǫ = m2lm2K, δ = m2πm2K,λ(x, y, z) = x2 + y2 + z2 −2xy −2yz −2zx,(48)and Γ is an overall normalization factor,Γ =GF√2 VudV ∗us2 α2m5K|g8|212π(4π)4 . (49)In our numerical estimates we shall use the following set of input values:⟨c¯qq⟩= −((190 −210) MeV)3; ⟨αsπ G2⟩= ((330 −390) MeV)4;Λχ = (700 −900) MeV; ΛMS = (100 −200) MeVandMQ = (250 −350) MeV.

(50)Then we have the following result for g8Re ω+, g8Re ωS, g8 (ω1 + 2 ω2) and g8 (ω1 −ω2),g8Re ω+ = 7.5+5−3;g8Re ωS = 5+4−2;g8 (ω1 + 2 ω2) = 0.12+0.02−0.01;g8 (ω1 −ω2) = −0.08+0.04−0.08,(51)where the central value corresponds to the input values ⟨c¯qq⟩= −(200 MeV)3,⟨αsπ G2⟩= (360 MeV)4, ΛMS = 150 MeV and MQ = 300 MeV with Λχ = 800MeV. Experimentally [15] we know that,g8Re ω+ = 4.6+1.2−0.7;g8 (ω1 + 2 ω2) = 0.41+0.10−0.05.

(52)In view of the these experimental results a value of the quark condensate lower than−(210 MeV)3 turn out to be not favoured.Our results tell us that the combination of counterterms for K+ →π+ l+l−,K0 →π0 l+l−and η →¯K0 l+l−decay amplitudes, ωS and ω+ [1], depend strongly13

on the penguin diagrams (both hadronic and electroweak). In addition, it turnsout that the coupling g8 ωS only depends on the penguin operators whereas g8 ω+depends also on the non-penguin Wilson coefficients.

This fact could be used formeasuring their respective strength.With the value of ωS in Eq. (51) we can predict the following branching ratio,Γ(K0S →π0e+e−) ≃2.6 × 10−9 [Re ω2S −0.66 Re ωS + 0.11]≃(0.5 −5) × 10−9,(53)for which there is an experimental upper bound of the order of 10−5 [17].

We can alsogive a prediction for the ratio of decay rates of the K+ →π+e+e−and K0S →π0e+e−transitions;Γ(K0S →π0e+e−)Γ(K+ →π+e+e−) ≃Re ω2S −0.66 Re ωS + 0.11Re ω2+ −0.59 Re ω+ + 0.09 = 0.30+0.50−0.25. (54)The coupling constant ˆc introduced in Eq.

(19) can also be determined fromthe results above. It turns out that there is no contribution from the electroweakpenguin Q7 to the ˆc coupling constant.

For its real part we findRe ˆc = −0.7 ± 0.5. (55)which translates in the following prediction for the branching ratio of the transitionK+ →π+γγ [1]:BR(K+ →π+γγ) = (5.2 ± 0.7) × 10−7.

(56)The experimental upper limit depends very much on the π+ energy spectrum, givinga wide range of allowed values [18],BR(K+ →π+γγ) ≤1.5 × 10−4. (57)The imaginary part of ˆc vanishes since there is no contribution of the electromag-netic penguin Cγ7 to this coupling.

This implies that there is no charge asymmetryΓ(K+ →π+γγ) −Γ(K−→π−γγ) from the CP-violating phase of the CKM mixingmatrix [1] appearing in the electromagnetic penguin Wilson coefficient.Finally, for the combinations ω1 + 2 ω2 −ω′1 and ω′2 introduced in Eq. (1) weget the results,14

g8 ( ω1 + 2 ω2 −ω′1) = 0.02 ± 0.01 ,ω′2 = 0 . (58)These two combinations turn out to be independent of the electroweak penguinoperator Q7 and therefore real.

They can be used in the theoretical prediction ofthe electric-type amplitude of the K+ →π+π0γ transitions, see Ref. [2].6ConclusionsIn the framework of the effective action aproach for four-quark operators [3], wehave calculated the various coupling constants that enter in the chiral perturbationtheory prediction for the K →πγ∗→πl+l−, K →πγγ and K →ππγ decay rates,[1, 2].

These constants are not determined by symmetry requirements alone. Theyturn out to depend strongly on the effective action of the penguin operators Q6 andQ7.We have given a prediction for the phenomenological constants ω+, ωS and ˆcdefined in Ref.

[1], which fix the decay rates for K →πγ∗and K+ →π−γγ. Inparticular we have predicted the branching ratio for K0S →π0e+e−in (53) and theratio of decay rates of K0S →π0e+e−and K+ →π+e+e−in (54).

We have giventhe branching ratio for K+ →π+γγ in (56) and found that there is no chargeasymmetry Γ(K+ →π+γγ) −Γ(K−→π−γγ) coming from the electromagneticpenguin Wilson coefficient Cγ7 . In the transition K →ππγ, further countertermsare possible.

They have been classified in Ref. [2].

The coupling constants thatmodulate these new counterterms ω′1 and ω′2 have also been predicted in (58).AcknowledgementsWe wish to thank Eduardo de Rafael for suggesting this calculation to us andfor many useful discussions. We have also benefited from discussions with GerhardEcker, Toni Pich and Josep Taron.

We would like to thank Lars H¨ornfeldt for hishelp with the algebraic manipulating program STENSOR. The work of one of us(J.P.) has been supported in part by CICYT, Spain, under Grant No.AEN90-0040.

J.P. is also indebted to the Spanish Ministerio de Educaci´on y Ciencia for afellowship.15

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